[Math] Developing methods for determining the balance and reliability of dice.

[Zogg]

Just ask your DM if he thinks the die is loaded. If your DM says, "No.", ask if you can trade d20s and d6s. I mean, if they aren't loaded, trading them won't hurt anyone, right? If your DM does not agree to this, protest.

If your DM says "Yes." and then laughs evilly I would run.

 

[Wolfspirit]

Heh, check out my "Statistically proving that I'm unlucky" thread here . I'm basically going the opposite direction that you are; I'm trying to show that it's not the dice that matter, my luck just sucks.

If you're trying to show that there's a statistical improbability due to a defect in the die, you're probably going to have to roll it at least a thousand times or so. Just rolling it twenty times and seeing the results wouldn't cut it.

(Oh, and not to derail the topic or anything, but last game we were fighting something that for the most part required a nat 20 to hit, which a few people were actually doing. I then looked at my research, and told them I hadn't rolled a 20 in 189 rolls)

 

[tarchon]

Heh, check out my "Statistically proving that I'm unlucky" thread here . I'm basically going the opposite direction that you are; I'm trying to show that it's not the dice that matter, my luck just sucks.

If you're trying to show that there's a statistical improbability due to a defect in the die, you're probably going to have to roll it at least a thousand times or so. Just rolling it twenty times and seeing the results wouldn't cut it.

It depends on how much confidence you want and what you want to test exactly. For example, testing the mean doesn't take that many rolls to get reasonable accuracy/confidence.

 

[Umbran]

For example, testing the mean doesn't take that many rolls to get reasonable accuracy/confidence.

Yes, but testing the mean doesn't tell you if the die is fair. If your d20 only rolls 20s and 1s, it'll show the same mean as a fair die.

 

[tarchon]

The first big problem I see a lot when people discuss random number generation is this term "fair" - what does it mean to you? Equal probability of outcomes, uncorrelated outcomes? The other big problem is the lack of specification for accuracy and confidence for the test.
Asking "is this die fair?" is like asking "is my car too green?" More information is needed to answer the question usefully.
Testing the mean gives you some information, and it's information you can get fairly accurately with a comparatively small number of trials. Whether it's a useful statistic depends on what you're doing with the numbers generated. With MdN stat generation, that's an excellent parameter to look at, since it is efficient to measure and you probably don't care much if 4 comes up a little more frequently than 5, as long as it's balanced out somewhere else. With thresholding mechanics, like d20 to hit, what's more important is the likelihood of each outcome, or more specifically, the equalities of the outcomes. That (usually) takes more trials to get accurately, however the test of the mean is also sensitive to defects in the frequencies of various outcomes. Whether it's sensitive enough depends on what accuracy you require and what kinds of "unfairness" you need to detect, which practically also depends on what kinds of "unfairness" are likely to occur as well as how significant they are. It makes little sense to design a test based on its ability to determine whether the die only rolls 1s and 20s - does that happen a lot? If you simply test outcome frequencies, that could also fail to detect a die that always rolls numbers in pairs, like 20, 20, 8, 8, 17,17..., however this mode of "unfairness" is similarly so unlikely that it's not worth designing a test around it. Now, people who gripe about computer PRNGs, to bring in a related FAQ, are griping about such a lack of independence between trials, which to me is usually so small in degree as to be irrelevant for game dice rolling, but that can be tested too.
Anything anybody wants to know about die outcome probability distributions can be tested with standard formulas to a known degree of accuracy and confidence, but you first have to know how to describe what you want to test.

 

[Umbran]

The first big problem I see a lot when people discuss random number generation is this term "fair" - what does it mean to you? Equal probability of outcomes, uncorrelated outcomes? The other big problem is the lack of specification for accuracy and confidence for the test.

In a statistical sense, "fair" is very well defined - equal probability of outcomes and uncorrelated results.

I note that if the outcomes are correleated, then in a major sense they are not of equal probability. Correlation implies unequal probabilities, at least on the short scale.

 

[Zander]

There's a professor in Canada who has invented a machine that rolls and reads dice. Granted, most of us, myself included, don't have access to such a machine. But at least in theory we can reach a high N without spending a great deal of time rolling dice.

tarchon is right: fairness is subjective. However, we can settle on conventions about how random a die should be to be considered "fair". It should be as random as casino dice. If that's too high a boundary than perhaps a lower boundary should be as random as a large sample of "shop" dice (ordinary d6's). The results of casino dice (and separately for shop dice) can be considered "expected" values for fair dice and then compared to test dice using Chi-square and other statistical methods.

 

[ichabod]

Myabe a good compromise test would be a double test of the mean and the standard deviation, using Bonferroni style confidence intervals. The mean would check the general spread, and the standard deviation would catch things like all 1's and 20's. I'll see if I can work out the theory later in the week, unless someone knows the distribution of the stardard deviation of a discrete uniform random variable.

 

[Umbran]

The mean would check the general spread, and the standard deviation would catch things like all 1's and 20's.

Well, the dsitribution for a fair single die should be completely flat. Standard deviations are intended to measure the width of a peak. The results of a d20 should have no peak whatsoever. So Standard deviation will probably fail to be a good measure.

 

[tarchon]

Myabe a good compromise test would be a double test of the mean and the standard deviation, using Bonferroni style confidence intervals. The mean would check the general spread, and the standard deviation would catch things like all 1's and 20's. I'll see if I can work out the theory later in the week, unless someone knows the distribution of the stardard deviation of a discrete uniform random variable.

I think it's
s^2=1/3 (n+1)(n-1) for 1dN
If you know the sum of squares and the sum for 1 to N, it works out pretty easily from the definition of the variance.